Three-dimensional numerical simulations of neutron star cores in the two-fluid MHD approximation: simple configurations

TL;DR

This study advances neutron star core magnetic-field modeling by performing 3D simulations of ambipolar diffusion in a two-fluid core (neutrons and charged particles) under the anelastic approximation, using Dedalus with simple poloidal-dipole and toroidal initial fields. Both the 1-barotropic and two-fluid models exhibit a five-stage evolution: initial acceleration, approach to a 2D force-balanced state, a non-axisymmetric instability, development of magnetic turbulence, and eventual resistive decay to small-scale structures; the two-fluid case shows slower acceleration and distinct mode content due to neutron coupling. A key result is that the instability develops on Alfvén timescales, not the ambipolar-diffusion timescale, and that viscosity plays a crucial role in stabilizing small-scale motions, shaping the turbulence cascade and energy dissipation. The findings have implications for understanding transient magnetic evolution in young neutron stars and demonstrate the viability and limitations of 3D two-fluid MHD models for NS cores.

Abstract

Magnetic field evolution in neutron star cores is not fully understood. We describe the field evolution both for one barotropic fluid as well as two collisionally coupled barotropic fluids with different density profiles using the anelastic approximation and the Navier-Stokes equations to simulate the evolution in three dimensions. In the one-fluid case, a single fluid describes the motion of the charged particles. In the two-fluid model, the neutral fluid is coupled to the electrically conductive fluid by collisions, the latter being dragged by the magnetic field. In this model, both fluids have distinct density profiles. This forces them to move at slightly different velocities, resulting in a relative motion between the two barotropic fluids -- ambipolar diffusion. We develop a code based on Dedalus and study the evolution of simple poloidal dipolar and toroidal magnetic fields. Previous 2D studies found that poloidal magnetic fields evolve towards a stable Grad-Shafranov equilibrium. In our 3D simulations we find an instability of the two-fluid system similar to the one in the barotropic fluid system. After the instability saturates, a highly non-linear Lorentz force introduces small-scale fluid motion that leads to turbulence, development of a cascade and significant, non-axially symmetric changes in the magnetic field configuration. Fluid viscosity plays an essential role in regularizing the small-scale fluid motion, providing an energy drain.

Figures (36)

• Figure 1: Dimensionless numerical radial profiles of number density for neutrons $n_\mathrm{n}$, charged particles $n_\mathrm{c}$, and interaction between protons and neutrons $\gamma_\mathrm{np}$. These curves correspond to polynomial fits to the HHJ equation of state. The solid lines show our actual numerical fit, while pale triangles are used to show the numerical values directly computed using the HHJ equation of state. Only $\gamma_\mathrm{np}$ deviates noticeably around $r/R_\mathrm{c} = 0.8$.
• Figure 2: Initial conditions for the magnetic field. Upper panels are for poloidal field, lower panels for toroidal field. We show the magnetic field (left column) as well as its Lorentz force (right column).
• Figure 3: Evolution of kinetic and magnetic energies for 1-barotropic-fluid MHD simulation. Left panel: total magnetic and kinetic energies. Right panel: non-axisymmetric part of magnetic ($E_\mathrm{m} - E_\mathrm{m}^\mathrm{s}$) and kinetic ($E_\mathrm{k} - E_\mathrm{k}^\mathrm{s}$) energies. We mark the following stages: I is the acceleration stage, II is evolution towards 2D force balance, III is instability, IV is magnetic turbulence, V is resistive decay. The red dashed line in the right panel shows exponential growth with timescale $t_\mathrm{Alf}/2$. We show in grey the results of axially symmetric simulations.
• Figure 4: Root-mean square forces during 1-barotropic-fluid simulation A.
• Figure 5: Meridional cuts showing force balance during the acceleration stage for 1-barotropic fluid.